on prime near-rings with generalized...

Hindawi Publishing CorporationInternational Journal of Mathematics and Mathematical SciencesVolume 2008, Article ID 490316, 5 pagesdoi:10.1155/2008/490316

Research ArticleOn Prime Near-Rings with Generalized Derivation

Howard E. Bell

Department of Mathematics, Faculty of Mathematics and Science, Brock University,St. Catharines, Ontario, Canada L2S 3A1

Correspondence should be addressed to Howard E. Bell, [email protected]

Received 29 November 2007; Accepted 25 February 2008

Recommended by Francois Goichot

Let N be a 3-prime 2-torsion-free zero-symmetric left near-ring with multiplicative center Z. Weprove that if N admits a nonzero generalized derivation f such that f(N) ⊆ Z, then N is acommutative ring. We also discuss some related properties.

Copyright q 2008 Howard E. Bell. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

1. Introduction

LetN be a zero-symmetric left near-ring, not necessarily with amultiplicative identity element;and letZ be its multiplicative center. DefineN to be 3-prime if for all a, b ∈ N\{0}, aNb /= {0};and callN 2-torsion-free if (N,+) has no elements of order 2. A derivation onN is an additiveendomorphism D of N such that D(xy) = xD(y) + D(x)y for all x, y ∈ N. A generalizedderivation f with associated derivation D is an additive endomorphism f : N → N such thatf(xy) = f(x)y + xD(y) for all x, y ∈ N. In the case of rings, generalized derivations havereceived significant attention in recent years.

In [1], we proved the following.

Theorem A. If N is 3-prime and 2-torsion-free and D is a derivation such that D2 = 0, then D = 0.

Theorem B. If N is a 3-prime 2-torsion-free near-ring which admits a nonzero derivation D for whichD(N) ⊆ Z, then N is a commutative ring.

Theorem C. If N is a 3-prime 2-torsion-free near-ring admitting a nonzero derivation D such thatD(x)D(y) = D(y)D(x) for all x, y ∈ N, then N is a commutative ring.

In this paper, we investigate possible analogs of these results, where D is replaced by ageneralized derivation f .

mailto:[email protected]

2 International Journal of Mathematics and Mathematical Sciences

We will need three easy lemmas.

Lemma 1.1 (see [1, Lemma 3]). Let N be a 3-prime near-ring.

(i) If z ∈ Z \ {0}, then z is not a zero divisor.

(ii) If Z \ {0} contains an element z such that z + z ∈ Z, then (N,+) is abelian.

(iii) If D is a nonzero derivation and x ∈ N is such that xD(N) = {0} or D(N)x = {0}, thenx = 0.

Lemma 1.2 (see [2, Proposition 1]). If N is an arbitrary near-ring and D is a derivation on N, thenD(xy) = D(x)y + xD(y) for all x, y ∈ N.

Lemma 1.3. Let N be an arbitrary near-ring and let f be a generalized derivation on N with associatedderivation D. Then

(f(a)b + aD(b)

)c = f(a)bc + aD(b)c ∀a, b, c ∈ N. (1.1)

Proof. Clearly f((ab)c) = f(ab)c+abD(c) = (f(a)b+aD(b))c+abD(c); and by using Lemma 1.2,we obtain f(a(bc)) = f(a)bc + aD(bc) = f(a)bc + aD(b)c + abD(c).

Comparing these two expressions for f(abc) gives the desired conclusion.

2. The main theorem

Our best result is an extension of Theorem B.

Theorem 2.1. Let N be a 3-prime 2-torsion-free near-ring. If N admits a nonzero generalized derivationf such that f(N) ⊆ Z, then N is a commutative ring.

In the proof of this theorem, as well as in a later proof, we make use of a further lemma.

Lemma 2.2. Let R be a 3-prime near-ring, and let f be a generalized derivation with associatedderivation D /= 0. If D(f(N)) = {0}, then f(D(N)) = {0}.

Proof. We are assuming that D(f(x)) = 0 for all x ∈ N. It follows that D(f(xy)) = D(f(x)y) +D(xD(y)) = 0 for all x, y ∈ N, that is,

f(x)D(y) +D(x)D(y) + xD2(y) = 0 ∀x, y ∈ N. (2.1)

Applying D again, we get

f(x)D2(y) +D2(x)D(y) +D(x)D2(y) +D(x)D2(y) + xD3(y) = 0 ∀x, y ∈ N. (2.2)

TakingD(y) instead of y in (2.1) gives f(x)D2(y)+D(x)D2(y)+xD3(y) = 0, hence (2.2) yields

D2(x)D(y) +D(x)D2(y) = 0 ∀ x, y ∈ N. (2.3)

Now, substitute D(x) for x in (2.1), obtaining f(D(x))D(y) + D2(x)D(y) + D(x)D2(y) = 0;and use (2.3) to conclude that f(D(x))D(y) = 0 for all x, y ∈ N. Thus, by Lemma 1.1(iii),f(D(x)) = 0 for all x ∈ N.

Howard E. Bell 3

Proof of Theorem 2.1. Since f /= 0, there exists x ∈ N such that 0 /= f(x) ∈ Z. Since f(x) + f(x) =f(x + x) ∈ Z, (N,+) is abelian by Lemma 1.1(ii). To complete the proof, we show that N ismultiplicatively commutative.

First, consider the caseD = 0, so that f(xy) = f(x)y ∈ Z for all x, y ∈ N. Then f(x)yw =wf(x)y, hence f(x)(yw − wy) = 0 for all x, y,w ∈ N. Choosing x such that f(x) /= 0 andinvoking Lemma 1.1(i), we get yw −wy = 0 for all y,w ∈ N.

Now assume that D /= 0, and let c ∈ Z \ {0}. Then f(xc) = f(x)c + xD(c) ∈ Z; therefore,(f(x)c + xD(c))y = y(f(x)c + xD(c)) for all x, y ∈ N, and by Lemma 1.3, we see that f(x)cy +xD(c)y = yf(x)c + yxD(c). Since both f(x) and D(c) are in Z, we have D(c)(xy − yx) = 0 forall x, y ∈ N, and provided that D(Z) /= {0}, we can conclude thatN is commutative.

Assume now that D /= 0 and D(Z) = {0}. In particular, D(f(x)) = 0 for all x ∈ N. Notethat for c ∈ N such that f(c) = 0, f(cx) = cD(x) ∈ Z; hence by Lemma 2.2, D(x)D(y) ∈ Z andD(y)D(x) ∈ Z for each x, y ∈ N. If one of these is 0, the other is a central element squaring to 0,hence is also 0. The remaining possibility is that D(x)D(y) and D(y)D(x) are nonzero centralelements, in which case D(x) is not a zero divisor. Thus D(x)D(x)D(y) = D(x)D(y)D(x)yields D(x)(D(x)D(y) − D(y)D(x)) = 0 = D(x)D(y) − D(y)D(x). Consequently, N iscommutative by Theorem C.

3. On Theorems A and C

Theorem C does not extend to generalized derivations, even if N is a ring. As in [3], considerthe ring H of real quaternions, and define f : H → H by f(x) = ix + xi. It is easy to checkthat f is a generalized derivation with associated derivation given by D(x) = xi − ix, and thatf(x)f(y) = f(y)f(x) for all x, y ∈ H.

TheoremA also does not extend to generalized derivations, as we see by lettingN be thering M2(F) of 2 × 2 matrices over a field F and letting f be defined by f(x) = e12x. However,we do have the following results.

Theorem 3.1. Let N be a 3-prime near-ring, and let f be a generalized derivation on N with associatedderivation D. If f2 = 0, then D3 = 0. Moreover, if N is 2-torsion-free, then D(Z) = {0}.

Proof. We have

f2(xy) = f(f(x)y + xD(y)) = f(x)D(y) + f(x)D(y) + xD2(y) = 0 ∀x, y ∈ N. (3.1)

Applying f to (3.1) gives

f(x)D2(y) + f(x)D2(y) + f(x)D2(y) + xD3(y) = 0 ∀x, y ∈ N. (3.2)

Substituting D(y) for y in (3.1) gives

f(x)D2(y) + f(x)D2(y) + xD3(y) = 0; (3.3)

Therefore, by (3.2) and (3.3),

f(x)D2(y) = 0 ∀x, y ∈ N. (3.4)

It now follows from (3.3) that xD3(y) = 0 for all x, y ∈ N; and since N is 3-prime, D3 = 0.

4 International Journal of Mathematics and Mathematical Sciences

Suppose now that N is 2-torsion-free and that D(Z) /= {0}, and let z ∈ Z be such thatD(z) /= 0. Then if x, y ∈ N and f(N)x = {0}, then f(yz)x = f(y)zx + yD(z)x = 0 = yD(z)x;and since N is 3-prime and D(z) is not a zero divisor, x = 0. It now follows from (3.4) thatD2 = 0 and hence by Theorem A that D = 0. But this contradicts our assumption thatD(Z) /= {0}, hence D(Z) = {0} as claimed.

Theorem 3.2. Let N be a 3-prime and 2-torsion-free near-ring with 1. If f is a generalized derivation onN such that f2 = 0 and f(1) ∈ Z, then f = 0.

Proof. Note that f(x) = f(1x) = f(1)x + 1D(x), so

f(x) = cx +D(x), c ∈ Z. (3.5)

If c = 0, then f = D and D2 = 0, so D = 0 by Theorem A and therefore f = 0.If c /= 0, then c is not a zero divisor, hence by (3.4) D2 = 0 and D = 0. But then f(x) = cx

and f2(x) = c2x = 0 for all x ∈ N. Since c2 is not a zero divisor, we get N = {0}—acontradiction. Thus, c = 0 and we are finished.

4. More on Theorem C

In [4], the author studied generalized derivations f with associated derivation D which havethe additional property that

f(xy) = D(x)y + xf(y) ∀x, y ∈ N. (∗)

Our final theorem, a weak generalization of Theorem C, was stated in [4]; but the proof givenwas not correct. (At one point, both left and right distributivity were assumed.) We now haveall the results required for a proof.

Theorem 4.1. Let N be a 3-prime 2-torsion-free near-ring which admits a generalized derivation f withnonzero associated derivation D such that f satisfies (∗). If f(x)f(y) = f(y)f(x) for all x, y ∈ N, thenN is a commutative ring.

Proof. It is correctly shown in [4] that (N,+) is abelian and either f(N) ⊆ Z orD(f(N)) = {0}.Hence, in view of Theorem 2.1, wemay assume thatD(f(N)) = 0 and therefore, by Lemma 2.2,that f(D(N)) = {0}. We calculate f(D(x)D(y)) in two ways. Using the defining property of f ,we obtain f(D(x)D(y)) = f(D(x))D(y) +D(x)D2(y) = D(x)D2(y); and using (∗), we obtainf(D(x)D(y)) = D2(x)D(y)+D(x)f(D(y)) = D2(x)D(y). Thus,D2(x)D(y) = D(x)D2(y) for allx, y ∈ N. But sinceD(f(N)) = {0}, (2.3) holds in this case as well; thereforeD2(x)D(y) = 0 forall x, y ∈ N, hence by Lemma 1.1(iii) D2 = 0. Thus, D = 0, contrary to our original hypothesis,so that the case D(f(N)) = {0} does not in fact occur.

Acknowledgment

This research is supported by the Natural Sciences and Engineering Research Council ofCanada, Grant no. 3961.

Howard E. Bell 5

References

[1] H. E. Bell and G. Mason, “On derivations in near-rings,” in Near-Rings and Near-Fields (Tubingen, 1985),G. Betsch, Ed., vol. 137 of North-Holland Mathematics Studies, pp. 31–35, North-Holland, Amsterdam,The Netherlands, 1987.

[2] X.-K.Wang, “Derivations in prime near-rings,” Proceedings of the AmericanMathematical Society, vol. 121,no. 2, pp. 361–366, 1994.

[3] H. E. Bell and N.-U. Rehman, “Generalized derivations with commutativity and anti-commutativityconditions,” Mathematical Journal of Okayama University, vol. 49, no. 1, pp. 139–147, 2007.

[4] O. Golbasi, “Notes on prime near-rings with generalized derivation,” Southeast Asian Bulletin of Math-ematics, vol. 30, no. 1, pp. 49–54, 2006.

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